Question: Simplify the following expression and state the condition under which the simplification is valid: $k = \dfrac{a^2 + 8a}{a^2 + 7a - 8}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{a^2 + 8a}{a^2 + 7a - 8} = \dfrac{(a)(a + 8)}{(a - 1)(a + 8)} $ Notice that the term $(a + 8)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(a + 8)$ gives: $k = \dfrac{a}{a - 1}$ Since we divided by $(a + 8)$, $a \neq -8$. $k = \dfrac{a}{a - 1}; \space a \neq -8$